Equilibrium states on the Cuntz-Pimsner algebras of self-similar actions

TL;DR

This paper studies equilibrium states (KMS states) on Cuntz-Pimsner algebras arising from self-similar group actions, revealing explicit descriptions of these states across different temperature regimes and applying results to notable self-similar groups.

Contribution

It provides a detailed analysis of KMS states on these algebras, including explicit formulas and conditions for uniqueness, especially for contracting groups.

Findings

KMS states above critical temperature are given by traces on the group algebra.

At critical temperature, KMS states factor through the Cuntz-Pimsner algebra.

For contracting groups, the Cuntz-Pimsner algebra has a unique KMS state.

Abstract

We consider a family of Cuntz-Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz-Pimsner algebra; if the self-similar group is contracting, then the Cuntz-Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.